Fast Multigrid Solvers for Transport with Forward-Peaked Scattering

该方案的主要目标是开发快速、高效和健壮的多重网格算法，用于求解高前向峰值散射条件下的线性Boltzmann输运方程。最近，该项目团队为一个模型问题开发了一种角度多重网格算法，该算法捕获了二维“平地”模型中散射的玻尔兹曼输运方程的基本特征。因此，本项目的研究目标是将这种方法扩展到一种高效、有效的方法，用于真实介质中的真实三维散射，并具有准确的离散化。将已经开发的技术扩展到三维的挑战包括改进离散化以允许不连续的系数，以及在感兴趣的区域需要局部网格细化。此外，项目团队将在二维和三维空间维度对这些算法的收敛进行理论分析，为设计这些针对真实散射内核的算法提供关键的洞察力。准确和高效的前向峰值散射模型在生物医学和核工程应用中都具有重要意义。这一体系描述了用于放射治疗某些癌症肿瘤的电子束的散射，以及反应堆物理学和天体物理学中带电粒子的传输。虽然人们对有效和准确的带电粒子输运建模算法的兴趣由来已久，但这个问题仍然构成了巨大的挑战。本文所考虑的方法为这一领域的研究提供了新的方向。这项拟议的工作直接为生物医学和核工程师和科学家提供了模拟工具。PI和共同PI在计算科学和数学生物学社区中的积极作用确保了所产生的算法的及时和广泛的传播。此外，该项目直接涉及一名博士生，该博士生由国际和平研究所和联合国际积极指导，为培养一个重要研究领域的早期职业科学家做出了贡献。

The primary goal of this proposal is to develop fast, efficient, and robust multigrid-based solvers for the solution of the linear Boltzmann-transport equation in the regime of highly forward-peaked scattering. Recently, the project team have developed an angular multigrid algorithm for a model problem that captures the essential features of the Boltzmann-transport equation for scattering in a two-dimensional "Flatland" model. The research goals of this project are, thus, to extend this approach to an efficient and effective method for true three-dimensional scattering, in realistic media, with accurate discretizations. Among the challenges of extending the already developed technique to three dimensions are improving the discretization to allow discontinuous coefficients, and local grid refinement needed in regions of interest. Additionally, the project team will investigate theoretical analysis of the convergence of these algorithms, in both two and three spatial dimensions, providing critical insight into the design of these algorithms for realistic scattering kernels.Accurate and efficient models of forward-peaked scattering are of significant interest in both biomedical and nuclear engineering applications. This regime describes the scattering of electron beams, used in radiation therapy for the treatment of certain cancerous tumors, as well as the transport of charged particles in reactor physics and astrophysics. While there is a long history of interest in efficient and accurate algorithms for modeling charged-particle transport, the problem still poses formidable challenges. The approach considered here offers a new direction for research in this area. The proposed work leads directly to simulation tools for biomedical and nuclear engineers and scientists. The active roles of the PI and co-PI in the computational science and mathematical biology communities ensure timely and widespread dissemination of the resulting algorithms. Furthermore, the project directly involves a doctoral student, who is actively mentored by the PI and co-PI, contributing to the training of an early career scientist in an important field of research.